This is the third paper in a series of papers adressing the characteristic gluing problem for the Einstein vacuum equations. We provide full details of our characteristic gluing (including the $10$ charges) of strongly asymptotically flat data to the data of a suitably chosen Kerr spacetime. The choice of the Kerr spacetime crucially relies on relating the $10$ charges to the ADM energy, linear momentum, angular momentum and the center-of-mass. As a corollary, we obtain an alternative proof of the Corvino-Schoen spacelike gluing construction for strongly asymptotically flat spacelike initial data.
In this paper we develop a new approach to the gluing problem in General Relativity, that is, the problem of matching two solutions of the Einstein equations along a spacelike or characteristic (null) hypersurface. In contrast to the previous constructions, the new perspective actively utilizes the nonlinearity of the constraint equations. As a result, we are able to remove the $10$-dimensional spaces of obstructions to the null and spacelike (asymptotically flat) gluing problems, previously known in the literature. In particular, we show that any asymptotically flat spacelike initial data can…
In this paper we introduce the characteristic gluing problem for the Einstein vacuum equations. We present a codimension-$10$ gluing construction for characteristic initial data which are close to the Minkowski data and we show that the $10$-dimensional obstruction space consists of gauge-invariant charges which are conserved by the linearized null constraint equations. By relating these $10$ charges to the ADM energy, linear momentum, angular momentum and the center-of-mass we prove that asymptotically flat data can be characteristically glued (including the $10$ charges) to the data of a sui…
We construct initial data sets which satisfy the vacuum constraint equa- tions of General Relativity with positive cosmologigal constant. More pre- silely, we deform initial data with ends asymptotic to Schwarzschild-de Sitter to obtain non-trivial initial data with exactly Kerr-de Sitter ends. The method is inspired from Corvino's gluing method. We obtain here a extension of a previous result for the time-symmetric case by Chruściel and Pollack.
We study the free data in the Fefferman-Graham expansion of asymptotically Einstein metrics with non-zero cosmological constant. We prove that if $\mathscr{I}$ is conformally flat, the rescaled Weyl tensor at $\mathscr{I}$ agrees up to a constant with the free data at $\mathscr{I}$ , namely the traceless part of the $n$-th order coefficient of the expansion. In the non-conformally flat case, the rescaled Weyl tensor is generically divergent at $\mathscr{I}$ but one can still extract the free data in terms of the difference of the Weyl tensors of suitably constructed metrics, in full generality…
We establish an optimal gluing construction for general relativistic initial data sets. The construction is optimal in two distinct ways. First, it applies to generic initial data sets and the required (generically satisfied) hypotheses are geometrically and physically natural. Secondly, the construction is completely local in the sense that the initial data is left unaltered on the complement of arbitrarily small neighborhoods of the points about which the gluing takes place. Using this construction we establish the existence of cosmological, maximal globally hyperbolic, vacuum space-times wi…
We present two methods to include the asymptotic domain of a background spacetime in null directions for numerical solutions of evolution equations so that both the radiation extraction problem and the outer boundary problem are solved. The first method is based on the geometric conformal approach, the second is a coordinate based approach. We apply these methods to the case of a massless scalar wave equation on a Kerr spacetime. Our methods are designed to allow existing codes to reach the radiative zone by including future null infinity in the computational domain with minor modifications. W…
We present a local gluing construction for general relativistic initial data sets. The method applies to generic initial data, in a sense which is made precise. In particular the trace of the extrinsic curvature is not assumed to be constant near the gluing points, which was the case for previous such constructions. No global conditions on the initial data sets such as compactness, completeness, or asymptotic conditions are imposed. As an application, we prove existence of spatially compact, maximal globally hyperbolic, vacuum space-times without any closed constant mean curvature spacelike hy…
We prove a strong localized gluing result for the general relativistic constraint equations (with or without cosmological constant) in $n\geq 3$ spatial dimensions. We glue an $ε$-rescaling of an asymptotically flat data set $(\hatγ,\hat k)$ into the neighborhood of a point $\mathfrak{p}\in X$ inside of another initial data set $(X,γ,k)$, under a local genericity condition (non-existence of KIDs) near $\mathfrak{p}$. As the scaling parameter $ε$ tends to $0$, the rescalings $\frac{x}ε$ of normal coordinates $x$ on $X$ around $\mathfrak{p}$ become asymptotically flat coordinates on the asymptot…
In this paper we argue that the well-known maximal extensions of the Kerr and Kerr-Newman spacetimes characterized by a specific gluing (on disks) of two asymptotically flat regions with ADM masses of opposite signs are physically inconsistent and actually non-analytic. We also discover a correct geometrical interpretation of the surface $r=0$, $t={\rm const}$ - a dicone in the case of the Kerr solution and a more sophisticated surface of non-zero Gaussian curvature in the case of the Kerr-Newman solution - which suggests that the problem of constructing the maximal analytic extensions for the…
Generalising an analysis of Corvino and Schoen, we study surjectivity properties of the constraint map in general relativity in a large class of weighted Sobolev spaces. As a corollary we prove several perturbation, gluing, and extension results: we show existence of non-trivial, singularity-free, vacuum space-times which are stationary in a neighborhood of $i^0$; for small perturbations of parity-covariant initial data sufficiently close to those for Minkowski space-time this leads to space-times with a smooth global Scri; we prove existence of initial data for many black holes which are exac…
Different classes of exact solutions for the BHT massive gravity theory are constructed and analyzed. We focus in the special case of the purely quadratic Lagrangian, whose field equations are irreducibly of fourth order and are known to admit asymptotically locally flat black holes endowed with gravitational hair. The first class corresponds to a Kerr-Schild deformation of Minkowski spacetime along a covariantly constant null vector. As in the case of General Relativity, the field equations linearize so that the solution can be easily shown to be described by four arbitrary functions of a sin…
